Theorem issubrg2 20563

Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)

Hypotheses

Ref	Expression
issubrg2.b	⊢ 𝐵 = (Base‘𝑅)
issubrg2.o	⊢ 1 = (1r‘𝑅)
issubrg2.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
issubrg2	⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   1 (𝑥,𝑦)

Proof of Theorem issubrg2

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgsubg 20548	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrg2.o	. . . 4 ⊢ 1 = (1r‘𝑅)
3	2	subrg1cl 20551	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
4		issubrg2.t	. . . . . 6 ⊢ · = (.r‘𝑅)
5	4	subrgmcl 20555	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
6	5	3expb 1121	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
7	6	ralrimivva 3181	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
8	1, 3, 7	3jca 1129	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9		simpl 482	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10		simpr1 1196	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11		eqid 2737	. . . . . . 7 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
12	11	subgbas 19100	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
13	10, 12	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
14		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
15	11, 14	ressplusg 17248	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
16	10, 15	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
17	11, 4	ressmulr 17264	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅 ↾s 𝐴)))
18	10, 17	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅 ↾s 𝐴)))
19	11	subggrp 19099	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (𝑅 ↾s 𝐴) ∈ Grp)
20	10, 19	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Grp)
21		simpr3 1198	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
22		oveq1 7368	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
23	22	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
24		oveq2 7369	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
25	24	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
26	23, 25	rspc2v 3576	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
27	21, 26	syl5com 31	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
28	27	3impib 1117	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
29		issubrg2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
30	29	subgss 19097	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ⊆ 𝐵)
31	10, 30	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
32	31	sseld 3921	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
33	31	sseld 3921	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
34	31	sseld 3921	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
35	32, 33, 34	3anim123d 1446	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
36	35	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
37	29, 4	ringass 20228	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
38	37	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
39	36, 38	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
40	29, 14, 4	ringdi 20236	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
41	40	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
42	36, 41	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
43	29, 14, 4	ringdir 20237	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
44	43	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
45	36, 44	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
46		simpr2 1197	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1 ∈ 𝐴)
47	32	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐵)
48	29, 4, 2	ringlidm 20244	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ 𝐵) → ( 1 · 𝑢) = 𝑢)
49	48	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐵) → ( 1 · 𝑢) = 𝑢)
50	47, 49	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → ( 1 · 𝑢) = 𝑢)
51	29, 4, 2	ringridm 20245	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ 𝐵) → (𝑢 · 1 ) = 𝑢)
52	51	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐵) → (𝑢 · 1 ) = 𝑢)
53	47, 52	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑢 · 1 ) = 𝑢)
54	13, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53	isringd 20266	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Ring)
55	31, 46	jca 511	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))
56	29, 2	issubrg 20542	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
57	9, 54, 55, 56	syl21anbrc 1346	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
58	57	ex 412	. 2 ⊢ (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
59	8, 58	impbid2 226	1 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ‘cfv 6493  (class class class)co 7361  Basecbs 17173   ↾_s cress 17194  +_gcplusg 17214  ._rcmulr 17215  Grpcgrp 18903  SubGrpcsubg 19090  1_rcur 20156  Ringcrg 20208  SubRingcsubrg 20540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906  df-minusg 18907  df-subg 19093  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-subrng 20517  df-subrg 20541

This theorem is referenced by:  opprsubrg  20564  subrgint  20566  issubrg3  20571  issubrgd  21179  cnsubrglem  21409  cnsubrglemOLD  21410  mplsubrg  21996  mplind  22061  dmatsrng  22479  scmatsrng  22498  scmatsrng1  22501  cpmatsrgpmat  22699  fxpsubrg  33253  elrgspnlem2  33322  constrsdrg  33938